How to recognize and solve numerically practical problems which may arise in your research. We will solve some serious problems using the full power of MATLAB's built in functions and routines. This class is geared for those who need to get the basics in scientific computing methods for data analysis. Many of today's major research methods for exploring data analysis will be covered: signal processing, frequency filtering, time-frencency analysis, wavelets, principal component analysis, proper orthogonal decomposition, empirical mode decomposition etc. Applications will range from image processing to characterizing atmospheric dynamics.

Syllabus

(1) Review of Statistics: (0 week)

We will begin with a brief review of statistical methods.
The principles of statistics will be largely applied
in a computational context for extracting meaningful information
from data.

(a) mean, variance, moments

(b) probability distributions

(c) significance testing, hypothesis testing

(2)
Spectral and Time-Frequency Analysis: (4 weeks)

We will introduce the ideas of signal processing, filtering,
time-frequency representations including wavelet expansions.
Our application will be largely to problems in image processing,
denoising and noise reduction.

(a)
digital signal processing

(b) noise reduction and filtering

(c) image processing and face recognition

(d) time-frequency methods and wavelets

(e) sparse representation and compressive sensing

(3) Dimensionality Reduction and Equation-Free Techniques: (6 weeks)

These methods are practical attempts to reduce the
dimensionality of the data as well as infer statistically
meaningful trends in what otherwise appears to be noisy data.

(a) Principal Component Analysis (PCA)

(b) Proper Orthogonal Decomposition (POD)

(c) Singular Value Decomposition (SVD)

(d) Dynamic Mode Decomposition (DMD)

(e) Model Reduction

(f) Multi-scale equation-free methods

(g) Clustering and classification

Grading

Your course grade will be determined entirely from
your homework (100%). There will be 6 homeworks over the quarter.

Each of the homework sets will be part of your final grade.
During the quarter, you will receive six homeworks
that you will turn in via the class DROPBOX. These six homeworks are
equally weighted and worth 100% of your grade.
This homework should be written as if it were an
article/tutorial being prepared for submission. I expect a high level of
professionalism on these reports. The following is the expected format for
homework submission:

Title/author/abstract Title,
author/address lines, and short (100 words or less) abstract. (It is
not to
be a separate title page!)

Sec. I. Introduction and Overview

Sec. II. Theoretical Background

Sec. III.
Algorithm Implementation and Development

Sec. IV.
Computational Results

Sec. V. Summary and Conclusions

Appendix A MATLAB functions used and brief implementation explanation

Appendix B MATLAB codes

Appendix C (optional) Any
algebraically intense calculations
(long and drawn out calculations have no
business in Sec. II!)

I will grade based upon how completely you solved the
homework as well as neatness and little things like: did you label your
graphs and include figure captions.

EACH HOMEWORK IS WORTH 10 POINTS. Five points will be given for
the overall layout, correctness and neatness of the report, and five
additional points will be for specific things that the TAs will look for in
the report itself. We will not tell you these things ahead of time as a
good and complete report should have them as part of the explanation of
what you did. For example, in the first homework, the TAs may look to see
if you talked about the fact that you must rescale the wavenumbers by
2*pi/L since the FFT assumes 2*pi periodic signals. This is a detail that
is important, so it would be expected you would have it. If you do, you
get the point, if not, then you miss a point.

NOTE: The report does not have to be long. But it does have to
be complete.

NOTE 2: This report is not for me, it is for you! Specifically,
for the future you. So write a nice report so that you could reproduce
the results if you need the methods addressed here in another year or more.

A few things should be kept in
mind when generating your reports:

1. Use a professional
grade word processor (Latex or MSword,
for example)

2. For
equations: Latex already does a nice job, but in
Word, use Microsoft
Equation Editor 3.0

3. Label your graphs. Include brief figure
captions.
Reference the figure in the text with a more detailed account of
the
figure.

4. Figures should be set flush with the top or
bottom of a page.

5. Label all equations.

6.
Provide references where appropriate.

7. All coding should be
shuffled to Appendix A and B.
Reference it when necessary.

8.
Always remember: this report is being written for YOU!
So be clear and
concise.